Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1. The title of the article is too long and fails to highlight the key innovations and research focuses.
2. The beginning of the abstract needs to briefly introduce the research background, raise the key scientific questions to be solved, and facilitate readers to quickly understand the purpose and thought process of the article. Additionally, avoid displaying research conclusions in large chunks, and highlight the main conclusions. Finally, provide some quantitative indicator results as much as possible to demonstrate the advancement and professionalism of the research.
3. The research applies Shannon entropy analysis to Polish climate variability, which has a certain degree of innovation. However, there is still room for improvement in terms of innovation. Compared with existing research, the unique contributions of this research in terms of analysis methods or conclusions are not prominent enough, and it is suggested to further compare and emphasize the innovative points of the research.
4. The literature review covers various aspects, including the application of entropy in climate research and Polish climate characteristics, which is relatively comprehensive. However, when citing literature, some literature is not interpreted in depth, and the connection and difference between existing research and this research are not fully demonstrated. For example, for the many studies mentioned, some key literature can be selected to analyze their research methods and conclusions in detail, and to illustrate the inspiration for this research.
5. The methodology section provides a detailed introduction to the data sources and analysis methods, including Copula functions and Bootstrap resampling technology, which is relatively complete and correct. However, the basis for method selection can be further strengthened. For instance, when choosing the Clayton copula function, in addition to explaining its applicability to extreme events, the limitations of other Copula functions in this research can be compared, making the method selection more convincing.
6. The data comes from NOAA's daily gridded data, which has a certain degree of authority. However, the article does not detail the quality control measures for the data, and how the consistency and accuracy of the data are guaranteed in time and space are not mentioned. It is suggested to supplement the content related to data quality control to enhance the reliability of the data.
7. The overall English grammar and vocabulary are basically accurate, but there are some minor problems. For example, some sentences are slightly verbose and can be optimized to make the expression more concise and clear. Additionally, some professional vocabulary is used at a high frequency, and it can be appropriately replaced to enrich the language of the article.
8. The discussion and analysis section combines the changes in Shannon entropy values to conduct an in-depth exploration of the temporal and spatial characteristics of climate variability, influencing factors, and provides suggestions for public management. However, when discussing the relationship between climate variability and global climate change, further in-depth analysis can be conducted, such as quantitative analysis of the correlation between the two, to make the discussion more in-depth.
9. The article does not clearly point out the deficiencies in the research. Any research may have certain limitations, such as data limitations and potential methodological biases. It is suggested to supplement the content of research deficiencies and explain the possible impact of these deficiencies on the research results.
10. The research mainly analyzes the climate in Poland, and the conclusions have a certain application value in Poland. However, the universality of the research conclusions in other regions is not discussed, and the climate characteristics of different regions vary greatly. The applicability of the research methods and conclusions in other regions needs further consideration. It is suggested to mention the applicable range of the research conclusions and possible extension directions in the conclusion section.

Comments on the Quality of English Language

The English could be improved to more clearly express the research.

Author Response

Responses to Reviewer 1

 

Response to Reviewer 1 Comments
1. Summary
Thank you for taking the time to review this manuscript. Below, you will find detailed responses to the comments, with the corresponding revisions and corrections highlighted or tracked in the re-submitted files.
2. Questions for General Evaluation	Reviewer’s Evaluation	Response and Revisions
Does the introduction provide sufficient background and include all relevant references?	Must be improved	Yes, the introduction provides sufficient background and includes all relevant references. Key prior studies on climate variability in Poland and Central Europe—such as those by Kundzewicz, Miętus, Cebulska, Twardosz, and Wałęga—have been appropriately cited. Additionally, the introduction outlines the limitations of traditional statistical approaches and clearly explains the rationale behind using Shannon entropy and copula-based modeling. Recent studies on entropy in climate analysis, including those by Rodrigues da Silva and Hao, are also referenced. To strengthen the context, a paragraph was added to highlight the research impetus, situating this study within both theoretical and applied frameworks. All reviewer concerns regarding novelty, methodological justification, and literature coverage have been addressed.
Is the research design appropriate?	Yes
Are the methods adequately described?	Must be improved	Yes, the methods are adequately described. The article clearly outlines each step of the methodological framework, including the selection and preprocessing of climate data, the estimation of marginal distributions using Maximum Likelihood Estimation (MLE), and the statistical validation of these distributions using the Anderson–Darling test. The choice of the Clayton copula is thoroughly justified in terms of its suitability for modeling lower-tail dependence, which is relevant for analyzing drought conditions.   Additionally, the computation of Shannon entropy using joint probability distributions derived from copula models is explained in detail, along with the application of bootstrap resampling and numerical integration to ensure robust estimation. The study also describes the use of the Mann–Kendall and Pettitt tests for trend detection, and phase-space reconstruction via delayed coordinates for dynamic system analysis.   Newly added sections further clarify the rationale for methodological choices, such as the Granger causality test applied to explore temporal relationships between local entropy and global indices (e.g., NINO3.4). All procedures are supported by appropriate references, and the revised text responds to previous concerns about transparency and reproducibility.
Are the results clearly presented?	Can be improved	Yes, the results are clearly presented, with detailed tables, figures, and explanations that highlight spatial, temporal, and statistical patterns in entropy. Key findings are well-visualized and supported by appropriate statistical evidence.
Are the conclusions supported by the results?	Can be improved	Yes, the conclusions are well supported by the results, which consistently demonstrate the observed trends in entropy and their links to climate variability.

 

 

Point-by-point response to Comments and Suggestions for Authors

1. The title of the article is too long and fails to highlight the key innovations and research focuses.

 

The Dynamics of Shannon Entropy in Analyzing Climate Variability for Modeling Temperature and Precipitation Uncertainty in Poland

 

 

1. The beginning of the abstract needs to briefly introduce the research background, raise the key scientific questions to be solved, and facilitate readers to quickly understand the purpose and thought process of the article. Additionally, avoid displaying research conclusions in large chunks, and highlight the main conclusions. Finally, provide some quantitative indicator results as much as possible to demonstrate the advancement and professionalism of the research.

 

The abstract text has been revised.

 

1. The research applies Shannon entropy analysis to Polish climate variability, which has a certain degree of innovation. However, there is still room for improvement in terms of innovation. Compared with existing research, the unique contributions of this research in terms of analysis methods or conclusions are not prominent enough, and it is suggested to further compare and emphasize the innovative points of the research.

 

The comment has been addressed, and all innovative insights and applied techniques have been highlighted in the text (including in the conclusion).

 

1. The literature review covers various aspects, including the application of entropy in climate research and Polish climate characteristics, which is relatively comprehensive. However, when citing literature, some literature is not interpreted in depth, and the connection and difference between existing research and this research are not fully demonstrated. For example, for the many studies mentioned, some key literature can be selected to analyze their research methods and conclusions in detail, and to illustrate the inspiration for this research.

 

A paragraph explaining the research impulse has been added.

Among the numerous previous studies on climate variability in Poland and Central Europe, particular importance was attributed to statistical analyses of temperature and precipitation trends, such as those conducted by Kundzewicz and Miętus [38], [24], as well as Cebulska and Twardosz [26], Wałęga and Młyński [34], [27]. The use of information entropy in climate analyses was also suggested in studies by Rodrigues da Silva and Hao [39], [31], [40], although it was not always combined with a copula-based approach. It was precisely the limitations of traditional methods and the lack of a framework for capturing complex dependencies between variables that formed the starting point for this study. The selection of the Clayton copula function, as a tool for modeling asymmetric dependencies between precipitation and temperature, was inspired by its applications in hydrological and financial analyses – particularly by Nelsen and Hao [40], [41], [42]. These studies demonstrated that modeling extremes and dependency structures requires moving beyond classical linear correlations. Moreover, the approach utilizing phase space and entropy trajectories was inspired by research on chaotic systems – especially by Lorenz [43], [44], and Silva and Bhattacharya [45], [46]. A detailed review of the methodologies in these studies enabled the development of an approach tailored to the conditions in Poland, taking into account both temporal and spatial dimensions. In this way, the present study extends and integrates several research perspectives, creating a new tool for assessing climate variability.

1. The methodology section provides a detailed introduction to the data sources and analysis methods, including Copula functions and Bootstrap resampling technology, which is relatively complete and correct. However, the basis for method selection can be further strengthened. For instance, when choosing the Clayton copula function, in addition to explaining its applicability to extreme events, the limitations of other Copula functions in this research can be compared, making the method selection more convincing.

 

The Clayton copula was selected due to its ability to model strong dependencies in the lower tail of the distribution, which is particularly important when analyzing phenomena such as droughts—extremely low precipitation combined with high temperatures. Alternative copula functions, such as the Gumbel or Frank copulas, are better suited for modeling upper-tail dependence or moderate correlations, making them less appropriate in the case of the asymmetric dependence relevant to this study. The Gaussian copula assumes symmetric dependence, which does not reflect the actual, often nonlinear, relationship between precipitation and temperature under Polish climatic conditions. Therefore, the Clayton copula provides the most suitable fit for the data, and its application is justified both theoretically and empirically.

 

1. The data comes from NOAA's daily gridded data, which has a certain degree of authority. However, the article does not detail the quality control measures for the data, and how the consistency and accuracy of the data are guaranteed in time and space are not mentioned. It is suggested to supplement the content related to data quality control to enhance the reliability of the data.

 

In the data section, the following comment was added:

To ensure high quality, consistency, and comparability of the data across time and space, this study used climate data from the NOAA database, which applies uniform quality control procedures. This dataset was chosen for its homogeneity—both precipitation and temperature data were developed at the same spatial resolution (0.25° × 0.25°), by the same institution, and using standardized interpolation algorithms. NOAA data are routinely validated for anomalies, gaps, and inconsistencies through statistical methods and cross-checked against ground-based and satellite observations [48], [49], [50], [51]. As a result, a reliable and stable dataset was obtained, providing a solid foundation for the long-term analysis of entropy trends.

 

 

1. The overall English grammar and vocabulary are basically accurate, but there are some minor problems. For example, some sentences are slightly verbose and can be optimized to make the expression more concise and clear. Additionally, some professional vocabulary is used at a high frequency, and it can be appropriately replaced to enrich the language of the article.

 

The terminology has been revised, and the sentences have been shortened, making them more concise and easier for the reader to understand.

 

 

1. The discussion and analysis section combines the changes in Shannon entropy values to conduct an in-depth exploration of the temporal and spatial characteristics of climate variability, influencing factors, and provides suggestions for public management. However, when discussing the relationship between climate variability and global climate change, further in-depth analysis can be conducted, such as quantitative analysis of the correlation between the two, to make the discussion more in-depth.

 

A section has been added: The Relationship Between Climate Variability and Global Climate Change Through the Lens of Shannon Entropy

To gain deeper insight into the relationship between local climate variability in Poland and global climate trends, an additional analysis was conducted to examine the correlation between Shannon entropy values and selected global variables, such as the global land-ocean surface temperature anomaly (NASA Global Mean Estimates – Land-Ocean Temperature Index) and the ENSO index (NINO3.4). For the period 1941–2010, a significant positive Pearson correlation was found between entropy and the global temperature increase: ρ = 0.650 (p < 0.05) in summer and ρ = 0.826 (p < 0.05) in winter, suggesting that intensifying climate change may contribute to increased weather instability in Poland. To investigate the potential direction of influence, the Granger causality test was applied to determine whether the NINO3.4 index could be considered a cause of changes in Shannon entropy. The analysis was conducted over a wide range of lags, from 1 to 24 months. For lags of 8, 9, and 11 months, the test showed statistical significance (e.g., p = 0.0453 for lag 9); however, at other lags, the results were not significant. F-values for most tests did not exceed the critical values at the 0.05 significance level. The occurrence of significance only at selected lags may indicate instability in the relationship over time or the presence of nonlinear and seasonal dependency mechanisms between global ENSO variability and local climate entropy. While these results do not provide strong evidence of clear Granger causality, they may suggest seasonally dependent, delayed teleconnection effects. Notably, periods of intense El Niño events (e.g., 1982–1983, 1997–1998) coincide with distinct jumps in entropy in Poland.

 

Fig. Variability of entropy and indices

 

Granger causality test

Lag	F-stat	p-value	Krit (alfa=0.05)
1	1.1714	0.2797	3.861
2	0.6633	0.5156	3.0147
3	1.1966	0.3105	2.6238
4	1.0482	0.3818	2.3909
5	0.8685	0.5021	2.2332
6	0.54	0.7779	2.118
7	0.6039	0.7529	2.0293
8	1.9731	0.0481	1.9584
9	1.9355	0.0453	1.9002
10	1.8043	0.0575	1.8513
11	1.8983	0.0375	1.8096
12	0.7758	0.6757	1.7735
13	0.7615	0.7012	1.7418
14	0.7704	0.7017	1.7137
15	0.8464	0.6255	1.6887
16	0.9021	0.5669	1.6662
17	1.0508	0.4011	1.6458
18	1.0927	0.3564	1.6272
19	1.0869	0.3611	1.6102
20	1.1106	0.3346	1.5945
21	1.0738	0.3734	1.5801
22	1.0801	0.3651	1.5667
23	1.1017	0.3392	1.5543
24	1.0936	0.347	1.5427

 

 

 

1. The article does not clearly point out the deficiencies in the research. Any research may have certain limitations, such as data limitations and potential methodological biases. It is suggested to supplement the content of research deficiencies and explain the possible impact of these deficiencies on the research results.

 

Despite the use of a well-established methodology, the study has certain limitations stemming from both the nature of the data and the adopted analytical assumptions. Although the NOAA data are characterized by high quality and spatial consistency, they do not fully capture local climatic conditions, especially in areas with highly diverse topography. The applied resolution of 0.25° × 0.25° may fail to reflect microscale atmospheric phenomena such as local convective storms or intense point precipitation. The statistical methods based on copula functions and marginal distributions assume specific properties of data distribution, which may lead to inaccuracies if these assumptions are violated. The choice of the Clayton copula, although justified for analyzing lower-tail dependence, may not capture all types of interdependence between precipitation and temperature. Furthermore, the omission of factors such as land cover changes or human activity limits the ability to fully interpret the observed entropy trends.

 

1. The research mainly analyzes the climate in Poland, and the conclusions have a certain application value in Poland. However, the universality of the research conclusions in other regions is not discussed, and the climate characteristics of different regions vary greatly. The applicability of the research methods and conclusions in other regions needs further consideration. It is suggested to mention the applicable range of the research conclusions and possible extension directions in the conclusion section.

 

The study focuses primarily on the analysis of climate variability in Poland, which enables a detailed capture of local dependencies between temperature and precipitation. The obtained results hold significant practical value in the national context, particularly for adaptive planning and hydrological risk management. However, it should be noted that regional climatic, topographic, and hydrological conditions have a substantial impact on the data structure and the nature of observed trends. Therefore, directly transferring the findings to other regions may be limited and requires consideration of local specificities. In particular, the applied marginal distributions and type of copula may not be optimal for other climates, such as Mediterranean, continental, or tropical. Nevertheless, the methodology based on Shannon entropy and dependence modeling using copula functions can be successfully applied in other studies. In this sense, the research contributes a universal methodological component that can be replicated with appropriate parameter adjustments. In the future, it would be worthwhile to extend the analysis to other areas of Central and Eastern Europe to assess the extent to which the observed entropy patterns are common across a broader regional context.

 

The English language has been corrected for grammar and style, as well as for scientific terminology. Long sentences have been shortened, making the text clearer and easier to understand.

 

Thank you for the valuable suggestion.

March, 26,  2025

Bernard TWARÓG

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

I like the article but it has to be carefully rewritten.

There are many unfinished references in this article.  For example, "Climate data, particularly temperature and precipitation, form the foundation for climate change analyses [xx]".   Several more.  Or, "The study utilized a joint distribution based on a copula, describing the combined 307 variability of mean monthly temperature (modeled with a normal distribution) and 308 monthly precipitation sums (modeled with a gamma distribution) [xx]".

References 6 and 7 are data analyses, albeit climate data. Nothing to do with entropy.

References 18 and 19 are obscure to the point inferred by the author, (this is a foundational point for the article).  Reference 20 has been cited somewhat out of context.  Please elaborate specific findings in these references to assert what the authors have asserted on page 2 about superiority of "methods".

Maybe I am missing something, but what exactly does Shanon "entropy" have to do with the time series analysis of the data? There is no boundary like the Bekenstein bounds that connects this local statistical spread in the article for entropy, information.   The author should review several published articles on entropy rate generation and climate change rate.  What does "entropy" have to do with the science of weather and extreme events, with a time component required to assert that climate changes and extreme events are increasing? 

A Gaussian (e.g., a random variable) distribution of course has the most significant entropy amongst all random variables of equal variance, or the maximum entropy distribution under constraints of the mean and variance (see my comment below about marginal distributions for multivariate analysis). Volume and microstates are related, but microstate change must be addressed. From a microscopic standpoint, entropy can be linked to the probabilistic features of the accessible microstates of a system, i.e., to the peculiarities of the corresponding phase space. When the distribution is skewed (for example, because of a potential gradient), the entropy changes from that of the normal distribution over a study volume. Each such distribution is associated with an entropy, clearly the author addresses this part).   Thus, an entropy deficit can be associated with a normal and skewed distribution. It is called relative entropy or the Kullback–Leibler divergence, which is not addressed.  

However, I fail to see what this article's distribution and time series analysis has to do with entropy (even information type).  

I have not checked the data. I cannot figure out the relevance of equation 1, except for assessing the probability (again, not clear of which climate event) for this distribution.  I think this article would benefit from the author taking a little more time over it.

The author needs to assess and equate the marginal distributions to employ the Copula method.  Perhaps this is implicit and even stated - but I could not find a statement asserting this part.

Author Response

Responses to Reviewer 2

 

 

Response to Reviewer 2 Comments
1. Summary
Thank you for taking the time to review this manuscript. Below, you will find detailed responses to the comments, with the corresponding revisions and corrections highlighted or tracked in the re-submitted files.
2. Questions for General Evaluation	Reviewer’s Evaluation	Response and Revisions
Does the introduction provide sufficient background and include all relevant references?	Must be improved	Yes, the introduction provides sufficient background and includes all relevant references. Key prior studies on climate variability in Poland—such as those by Kundzewicz, Miętus, Cebulska, Twardosz, and Wałęga—have been appropriately cited. Additionally, the introduction outlines the limitations of traditional statistical approaches and clearly explains the rationale behind using Shannon entropy and copula-based modeling. Recent studies on entropy in climate analysis, including those by Rodrigues da Silva and Hao, are also referenced. To strengthen the context, a paragraph was added to highlight the research impetus, situating this study within both theoretical and applied frameworks. All reviewer concerns regarding novelty, methodological justification, and literature coverage have been addressed.
Is the research design appropriate?	Must be improved	Yes, the research design is appropriate and well-aligned with the study's objectives. The use of Shannon entropy as a quantitative measure of climate variability is justified, particularly for capturing changes in uncertainty and complexity that traditional statistical methods may overlook. The study employs a robust methodological framework, including the use of copula functions to model joint distributions of temperature and precipitation, bootstrapping for uncertainty estimation, and trend analysis using Mann-Kendall and Pettitt tests. The selection of the Clayton copula is theoretically motivated and empirically validated. Additionally, the integration of phase space analysis provides insight into the dynamical behavior of the climate system. The use of high-resolution NOAA data ensures spatial and temporal consistency.
Are the methods adequately described?	Must be improved	Yes, the methods are adequately described. The article clearly outlines each step of the methodological framework, including the selection and preprocessing of climate data, the estimation of marginal distributions using Maximum Likelihood Estimation (MLE), and the statistical validation of these distributions using the Anderson–Darling test. The choice of the Clayton copula is thoroughly justified in terms of its suitability for modeling lower-tail dependence, which is relevant for analyzing drought conditions.   Additionally, the computation of Shannon entropy using joint probability distributions derived from copula models is explained in detail, along with the application of bootstrap resampling and numerical integration to ensure robust estimation. The study also describes the use of the Mann–Kendall and Pettitt tests for trend detection, and phase-space reconstruction via delayed coordinates for dynamic system analysis.   Newly added sections further clarify the rationale for methodological choices, such as the Granger causality test applied to explore temporal relationships between local entropy and global indices (e.g., NINO3.4). All procedures are supported by appropriate references, and the revised text responds to previous concerns about transparency and reproducibility.
Are the results clearly presented?	Can be improved	Yes, the results are clearly presented, with detailed tables, figures, and explanations that highlight spatial, temporal, and statistical patterns in entropy. Key findings are well-visualized and supported by appropriate statistical evidence.
Are the conclusions supported by the results?	Must be improved	Yes, the conclusions are well-supported by the results presented in the study. The analysis demonstrates statistically significant trends in Shannon entropy over time, particularly highlighting increases in entropy during the summer and winter months, which suggests growing unpredictability in climate patterns. These findings are quantitatively backed by the use of bootstrap-resampled entropy values and validated through non-parametric statistical tests such as the Mann-Kendall and Pettitt tests, which confirm both monotonic and abrupt changes in trend behavior. The spatial differentiation of entropy across watersheds and administrative regions reinforces the conclusion that regional climate variability in Poland has intensified, and that this variability is not uniformly distributed. The incorporation of copula functions to model joint distributions of temperature and precipitation adds depth to the understanding of inter-variable dependencies, particularly in the tails of the distribution—critical for analyzing extremes like droughts or intense precipitation. The study further strengthens its conclusions by correlating entropy trends with global indices such as the Land-Ocean Temperature Index and ENSO (NINO3.4), providing evidence of external teleconnections. Although the Granger causality test yielded mixed results—showing statistical significance only at selected lags—this nuance is clearly acknowledged and interpreted cautiously. The study’s conclusions regarding the increasing complexity and instability of the climate system are supported by phase-space analysis, which visualizes transitions from stable to chaotic states in the entropy dynamics. The conclusions are not only consistent with the empirical results but also appropriately tempered by methodological limitations, and they offer practical implications for climate risk assessment and policy planning in Poland.

 

 

Point-by-point response to Comments and Suggestions for Authors

 

 

 

1. I like the article but it has to be carefully rewritten.

Thank you very much for your feedback. I appreciate your positive assessment and fully acknowledge the need for careful rewriting. The manuscript has been thoroughly revised to improve clarity, structure, and scientific precision.

 

2. There are many unfinished references in this article.  For example, "Climate data, particularly temperature and precipitation, form the foundation for climate change analyses [xx]".   Several more.  Or, "The study utilized a joint distribution based on a copula, describing the combined 307 variability of mean monthly temperature (modeled with a normal distribution) and 308 monthly precipitation sums (modeled with a gamma distribution) [xx]".

All "blind" references have been corrected and updated — I apologize for the oversight.

 

3. References 6 and 7 are data analyses, albeit climate data. Nothing to do with entropy.

 

[6] refers to information entropy and its application in climate analysis:
M. Atieh, R. Rudra, and B. Gharabaghi, ‘Investigation of spatial and temporal variability of precipitation using an entropy theory’, ASABE 1st Climate Change Symposium: Adaptation and Mitigation, pp. 191–194, 2015, doi: 10.13031/cc.20152141843.

Meanwhile, [7] referred to climate data:
World Meteorological Organization, Guide to Climatological Practices, 2018 edition, no. WMO-No. 100. 2018.

More appropriate references have been inserted and this has been revised in the updated version:
[7] V. de P. Rodrigues da Silva, A. F. Belo Filho, R. S. Rodrigues Almeida, R. M. de Holanda, and J. H. B. da Cunha Campos, ‘Shannon information entropy for assessing space-time variability of rainfall and streamflow in semiarid region’, Science of The Total Environment, vol. 544, pp. 330–338, 2016, doi: 10.1016/j.scitotenv.2015.11.082.

 

4. References 18 and 19 are obscure to the point inferred by the author, (this is a foundational point for the article).  Reference 20 has been cited somewhat out of context.  Please elaborate specific findings in these references to assert what the authors have asserted on page 2 about superiority of "methods".

 

Previous version:
[18] P. Singh, A. Gupta, and M. Singh, ‘Hydrological inferences from watershed analysis for water resource management using remote sensing and GIS techniques’, Egyptian Journal of Remote Sensing and Space Science, vol. 17, no. 2, pp. 111–121, 2014, doi: 10.1016/j.ejrs.2014.09.003.
[19] P. N. Lal et al., National systems for managing the risks from climate extremes and disasters, vol. 9781107025. 2012.
[20] R. K. Guntu and A. Agarwal, ‘Investigation of Precipitation Variability and Extremes Using Information Theory’, p. 14, 2021, doi: 10.3390/ecas2020-08115.

Has been revised to:
[18] Z. Hao, ‘Application of Entropy Theory in Hydrologic Analysis and Simulation’, Pengaruh Penggunaan PASTA LABU KUNING (Cucurbita Moschata) UNTUK SUBSTITUSI TEPUNG TERIGU DENGAN PENAMBAHAN TEPUNG ANGKAK DALAM PEMBUATAN MIE KERING, vol. 15, no. 1, pp. 165–175, 2016, [Online]. Available: https://core.ac.uk/download/pdf/196255896.pdf.
[19] S. Saha and S. Chattopadhyay, ‘Exploring of the summer monsoon rainfall around the Himalayas in time domain through maximization of Shannon entropy’, Theoretical and Applied Climatology, vol. 141, no. 1–2, pp. 133–141, 2020, doi: 10.1007/s00704-020-03186-4.

 

 

5. Maybe I am missing something, but what exactly does Shanon "entropy" have to do with the time series analysis of the data? There is no boundary like the Bekenstein bounds that connects this local statistical spread in the article for entropy, information.   The author should review several published articles on entropy rate generation and climate change rate.  What does "entropy" have to do with the science of weather and extreme events, with a time component required to assert that climate changes and extreme events are increasing? 

Shannon entropy, although originally derived from information theory, is widely applied in environmental and climate data analysis as a measure of uncertainty, disorder, and variability in probability distributions. In the context of time series of meteorological data (such as precipitation or temperature), information entropy is not used for dynamic time-based modeling (e.g., ARIMA models), but rather as a way to characterize the degree of unpredictability of variables within given periods (e.g., monthly time windows). It does not relate directly to limits such as the Bekenstein bound, as entropy in this context is not used in the physical (thermodynamic or quantum) sense, but rather in the statistical sense—as an indicator of complexity and dispersion of information in empirical data. In climate literature, Shannon entropy has been used in studies assessing seasonal variability, spatial comparisons, and for detecting changes in climate regimes.

Shannon entropy, although it does not directly account for time, can be used in time series analysis by calculating its values for successive periods (e.g., months or years). In this way, it helps capture changes in the unpredictability of temperature and precipitation distributions, making it a useful tool for studying climate variability and the intensification of extreme weather events. An increase in entropy over time may indicate growing complexity and irregularity of the climate—features often associated with the occurrence of extremes.

Variability and unpredictability (i.e., high entropy) are phenomena strongly linked to weather extremes. If data for a given month show high dispersion (e.g., ranging from very dry to very wet days), then:

• Entropy increases,
• The probability of extreme values (e.g., very intense rainfall) also increases.

Thus, entropy serves as a proxy (indirect indicator) of climate variability and extremeness.

 

6. A Gaussian (e.g., a random variable) distribution of course has the most significant entropy amongst all random variables of equal variance, or the maximum entropy distribution under constraints of the mean and variance (see my comment below about marginal distributions for multivariate analysis). Volume and microstates are related, but microstate change must be addressed. From a microscopic standpoint, entropy can be linked to the probabilistic features of the accessible microstates of a system, i.e., to the peculiarities of the corresponding phase space. When the distribution is skewed (for example, because of a potential gradient), the entropy changes from that of the normal distribution over a study volume. Each such distribution is associated with an entropy, clearly the author addresses this part).   Thus, an entropy deficit can be associated with a normal and skewed distribution. It is called relative entropy or the Kullback–Leibler divergence, which is not addressed.  

 

You're absolutely right — the Gaussian (normal) distribution indeed maximizes differential entropy among all continuous distributions with a given variance. This is a classical result from information theory and the theory of differential entropy. It means that any deviation from normality (e.g., skewness or kurtosis) implies an entropy deficit compared to the maximum entropy that would be achieved by a normal distribution with the same variance. In this context, an observed decline in entropy may indicate greater structure, lower randomness, or simply the presence of skewed or heavy-tailed distributions. Thus, Shannon entropy, when applied in climatological analyses, is sensitive not only to the "spread" of data but also to the shape of the distribution. I also agree that in comparing distributions (e.g., observed vs. reference, such as a normal distribution), a more informative measure could be relative entropy, or Kullback–Leibler (KL) divergence, which quantifies how much a given distribution diverges from a reference one.

Using KL divergence could allow us not only to measure general uncertainty (as with Shannon entropy), but also to identify how far local climate patterns deviate from some "theoretical" or historical normality. However, assuming a specific reference distribution (e.g., normal) without justification may be arbitrary and potentially misleading. KL divergence is only meaningful when:

• We know or justify that a particular distribution is the baseline — for example, “pre-climate change conditions,” or the distribution from a reference period like 1961–1990;
• We are interested in how the distribution changes over time, comparing distributions between time periods, not against a fixed theoretical model.

Therefore, a more appropriate approach is to measure the variability of the empirical distribution itself over time — and this is exactly where Shannon entropy functions as a non-theoretical, data-driven measure of uncertainty.

 

This clarification has been included in the article.

In this study, Shannon entropy was used as a measure of overall uncertainty and dispersion in monthly temperature and precipitation distributions without assuming any specific distributional shape (e.g., normality). While more advanced methods like Kullback–Leibler divergence exist in information theory, their use requires adopting a reference distribution or baseline, which may be difficult to define clearly in the case of complex and nonlinear climate processes. Instead, we focused on the empirical evolution of entropy over time and space as an indicator of local climate variability. This approach enables the detection of increasing climatic unpredictability without needing an "ideal" or predefined reference distribution, which is not always known or appropriate.

 

7. However, I fail to see what this article's distribution and time series analysis has to do with entropy (even information type).  

In the article, Shannon entropy was not applied to describe the temporal structure of the data in a strict sense, but rather to analyze monthly temperature and precipitation values in a cross-sectional manner. Its connection to time series lies in the fact that we analyze how the entropy of these distributions evolves over time — that is, from each monthly sample we construct an empirical distribution, calculate its entropy, and then analyze the resulting entropy time series for trends, change points, etc. The goal is therefore not classical time series modeling (e.g., ARIMA or GARCH), but rather the analysis of how probability distributions vary over time. This approach aligns with the growing interest in non-parametric measures of climate variability, where entropy is used as an indicator of the level of disorder or complexity in weather-related distributions. From an information-theoretic perspective, Shannon entropy quantifies the level of uncertainty — thus, an increase in entropy implies greater randomness in the distribution of climate variables. This has direct implications for forecasting difficulty and potentially higher susceptibility to extreme weather events.

 

8. I have notchecked the data. I cannot figure out the relevance of equation 1, except for assessing the probability (again, not clear of which climate event) for this distribution.  I think this article would benefit from the author taking a little more time over it.

 

The equations labeled as 1

,  , month=1,…,12  ,  , month=1,…,12

(1)

 describe specific segments of meteorological data (precipitation and temperature) in a seasonal framework, extracted for monthly analyses using a moving window approach. The months, of course, range from 1 to 12. The equation illustrates the preparation of a set of samples (data vectors), from which — for each month — a probability distribution is calculated, and based on that distribution, the Shannon entropy is then computed. This method of constructing samples enables tracking how the distribution of a given variable for a specific month changes over time, with the analysis window being shifted forward by one year (a so-called rolling window analysis). Analyzing the changes in distribution — not just in terms of means or variances — makes it possible to capture the unpredictability and instability of the climate, with Shannon entropy serving as a measure of that uncertainty. Thus, the focus is not on a single “point-in-time” distribution, but rather on its temporal evolution, which is key to understanding climate variability.

 

9. The author needs to assess and equate the marginal distributions to employ the Copula method.  Perhaps this is implicit and even stated - but I could not find a statement asserting this part.

 

The article applies a standard procedure for fitting marginal distributions for temperature and precipitation using maximum likelihood estimation (MLE), and their adequacy was evaluated using the Anderson-Darling test (ADT) at a 5% significance level. The selection of distributions was based on data characteristics; several candidates were considered, including GEV, log-normal, Weibull, gamma, and Gumbel distributions. The normal distribution was ultimately selected for temperature (validated by ADT), and the gamma distribution for precipitation (also validated by ADT). Based on these results, several copula functions were analyzed to construct the joint distribution: Clayton, Frank, Gumbel, Gaussian, and t-Copula. The best and comparable results were obtained using the t-Copula and Clayton copula. Ultimately, the Clayton copula was selected due to its ability to model strong dependencies in the lower tail of the distribution, which is particularly relevant for analyzing events such as droughts—i.e., extremely low precipitation accompanied by high temperatures. The copula parameter was optimized by fitting the theoretical cumulative distribution function (CDF) to the empirical one using the mean squared error (MSE) method. The theoretical CDF was constructed based on the marginal normal and gamma distributions with parameters estimated via MLE. This approach ensures methodological consistency and enables accurate representation of the interdependence between climatic variables.

 

The English language has been corrected for grammar and style, as well as for scientific terminology. Long sentences have been shortened, making the text clearer and easier to understand.

 

Thank you for the valuable suggestion.

March, 26,  2025

Bernard TWARÓG

 

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Clearly, the author has demonstrated clarity and understanding of the statistical issues.

Author Response

Responses to Reviewer

 

Response to Reviewer 1 Comments
1. Summary
Thank you for taking the time to review this manuscript. Below, you will find detailed responses to the comments, with the corresponding revisions and corrections highlighted or tracked in the re-submitted files.
2. Questions for General Evaluation	Reviewer’s Evaluation	Response and Revisions
Does the introduction provide sufficient background and include all relevant references?	Can be improved	Thank you for your valuable comment regarding the introductory section.  Relevant references were reviewed, updated, and expanded to ensure the section offers a more complete and coherent background for the study. The introduction has also been refined in terms of structure and content to improve clarity and contextual relevance.
Is the research design appropriate?	Yes
Are the methods adequately described?	Yes
Are the results clearly presented?	Yes
Are the conclusions supported by the results?	Not applicable	Thank you for your comment regarding the alignment between the conclusions and the presented results. We would like to kindly clarify that the conclusion section has been thoroughly revised and improved in terms of scientific rigor to ensure that it is fully supported by the analytical findings. Specifically, the conclusions are directly based on the quantitative outcomes reported throughout the study. The conclusions also reflect the results of the Granger causality test and the phase space trajectory analysis, which are clearly presented and interpreted in the main body of the manuscript. As such, the final section maintains a consistent and evidence-based connection with the core analytical work.. We hope that these revisions address the concern and confirm the coherence and methodological robustness of the conclusions.

 

 

Point-by-point response to Comments and Suggestions for Authors

1. Clearly, the author has demonstrated clarity and understanding of the statistical issues.

Thank you very much for your kind and encouraging remark:

“Clearly, the author has demonstrated clarity and understanding of the statistical issues.”

Your feedback has been truly motivating and deeply appreciated.

I would also like to express my sincere gratitude for the constructive and critical nature of your comments throughout the review process. They have significantly contributed to the scientific depth and overall quality of the work. Thanks to this thoughtful guidance, the final version of the manuscript has become more rigorous, coherent, and valuable from a methodological perspective.

Once again, thank you for your time and insightful observations.

Thank you for the valuable suggestion.

April, 03,  2025

Bernard TWARÓG

 

Author Response File: Author Response.pdf